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Question #59434

Difficulty: 8/10 | 🟢 Free

The Question

Let $n \in \mathbb{Z}^+$ and $a = \frac{2n^2 - 2n + 1}{2n}$ . Consider the function $f(x) = |[x]^2 - a x|$ , where $[ \cdot ]$ denotes the greatest integer function. Let $L$ and $R$ denote the left-hand and right-hand derivatives of $f(x)$ at $x = n$ , respectively. Let $P$ be the proposition ' $f(x)$ is continuous at $x=n$ ' and $Q$ be the proposition ' $f(x)$ is differentiable at $x=n$ '. Which of the following logical compound statements is true?

Options

A.
P \land \neg Q \land (L + R = 0)
B.
P \land Q \land (L - R = 0)
C.
\neg P \land \neg Q \land (L \cdot R = a^2)
D.
P \land \neg Q \land (L + R = 2a)

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Right-Hand vs Left-Hand Derivatives|Greatest Integer Function Traps|Modulus Sign Dependency

Why This Question Is Tricky

Examiner Levers

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the 'Lethal Collision' between a Greatest Integer Function (GIF) jump discontinuity and a modulus sign flip.

Students often blindly apply derivative formulas, incorrectly assuming the derivative of $[x]$ is zero everywhere, which leads to $L=R=-a$ .

The trap lies in the fact that as $x \to n^-$ , the GIF drops to $n-1$ , causing the internal expression of the modulus to become negative.

The modulus then flips the sign of the entire expression to maintain continuity, which inverts the slope.

Consequently, the left-hand derivative becomes $+a$ while the right-hand derivative is $-a$ , proving non-differentiability despite continuity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Right-Hand vs Left-Hand Derivatives
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free